Comments for CLosure

Like my teacher drew a function box than placed a "w" and just added a + sign in the middle than wrote Closed????

confused much

The answer:

I think I can give you a good working example of closure.

There are two questions you must answer for all problems involving closure:

1) What set of numbers is in use?

All real numbers? Or some real numbers?

All rational numbers? Or some rational numbers?

Whole numbers? Some whole numbers?

Integers?

…etc.

2) Closure WITH RESPECT TO what? When speaking of closure, it’s not just the word closure by itself.

Closure always means: Closure WITH RESPECT TO a mathematical operation.

Closure with respect to addition.
Closure with respect to subtraction.
Closure with respect to multiplication.
Closure with respect to division.

…and so on.

Since your teacher included a + sign in the question, I presume he or she is asking you about closure with respect to addition.

A practical example:

Suppose you go to a grocery store and buy two items. You’re going to use REAL NUMBERS to calculate the total cost of your purchases.

Item 1 costs $4.51
Item 2 costs $0.71

How much money do you need to buy these two items?
Just add the two numbers: $4.51 + $0.71 = $5.22

Notice that $4.51 is part of the set of all real numbers, and $0.71 is part of the set of all real numbers . . . but the addition of the two real numbers produces ANOTHER REAL NUMBER – it is also part of the set of all real numbers.

Therefore, the SET OF REAL NUMBERS is CLOSED UNDER ADDITION. When you add any two real numbers, you get another real number.

You are probably asking yourself why this is important.

It is important because you know in advance (before you add the two numbers) that the answer will be another real number – something you can use to understand how much money you are spending.

Suppose the set of real numbers was not closed under addition? Suppose the sum produced an imaginary number. If that were true you could never depend upon addition to produce a useful answer. Addition would not be practical – why bother adding anything – the result couldn’t be used.

Since your teacher used a function box, I presume it looked something like this:

x [input] >>>> W [in function box] >>>> W(x) [output]

I can’t be sure where the + sign appeared from your description.

However, I think the example I gave you will help you answer the question.